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Knot polynomials in the first non-symmetric representation. (English) Zbl 1285.81035
Summary: We describe the explicit form and the hidden structure of the answer for the HOMFLY polynomial for the figure-8 and some other 3-strand knots in representation [A.Mironov, A.Morozov and S.Natanzon, Theor. Math. Phys. 166, No. 1, 1–22 (2011; Zbl 1312.81125); translation from Teor. Mat. Fiz. 166, No. 1, 3-27 (2011)]. This is the first result for non-torus knots beyond (anti)symmetric representations, and its evaluation is far more complicated. We provide a whole variety of different arguments, allowing one to guess the answer for the figure-8 knot, which can be also partly used in more complicated situations. Finally we report the result of exact calculation for figure-8 and some other 3-strand knots based on the previously developed sophisticated technique of multi-strand calculations. We also discuss a formula for the superpolynomial in representation [loc.cit.] for the figure-8 knot, which heavily relies on the conjectural form of superpolynomial expansion nearby the special polynomial point. Generalizations and details will be presented elsewhere.

MSC:
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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